Integral of sin(4x-3)cos(x+5) dx
The solution
Detail solution
Rewrite the integrand:
sin ( 4 x − 3 ) cos ( x + 5 ) = 8 sin ( 5 ) sin 4 ( x ) cos ( 3 ) cos ( x ) − 8 sin 3 ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) − 4 sin ( 5 ) sin 2 ( x ) cos ( 3 ) cos ( x ) + 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 4 ( x ) + 4 sin ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) − 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 2 ( x ) + sin ( 3 ) sin ( 5 ) sin ( x ) − 8 sin ( 3 ) cos ( 5 ) cos 5 ( x ) + 8 sin ( 3 ) cos ( 5 ) cos 3 ( x ) − sin ( 3 ) cos ( 5 ) cos ( x ) \sin{\left(4 x - 3 \right)} \cos{\left(x + 5 \right)} = 8 \sin{\left(5 \right)} \sin^{4}{\left(x \right)} \cos{\left(3 \right)} \cos{\left(x \right)} - 8 \sin^{3}{\left(x \right)} \cos{\left(3 \right)} \cos{\left(5 \right)} \cos^{2}{\left(x \right)} - 4 \sin{\left(5 \right)} \sin^{2}{\left(x \right)} \cos{\left(3 \right)} \cos{\left(x \right)} + 8 \sin{\left(3 \right)} \sin{\left(5 \right)} \sin{\left(x \right)} \cos^{4}{\left(x \right)} + 4 \sin{\left(x \right)} \cos{\left(3 \right)} \cos{\left(5 \right)} \cos^{2}{\left(x \right)} - 8 \sin{\left(3 \right)} \sin{\left(5 \right)} \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \sin{\left(3 \right)} \sin{\left(5 \right)} \sin{\left(x \right)} - 8 \sin{\left(3 \right)} \cos{\left(5 \right)} \cos^{5}{\left(x \right)} + 8 \sin{\left(3 \right)} \cos{\left(5 \right)} \cos^{3}{\left(x \right)} - \sin{\left(3 \right)} \cos{\left(5 \right)} \cos{\left(x \right)} sin ( 4 x − 3 ) cos ( x + 5 ) = 8 sin ( 5 ) sin 4 ( x ) cos ( 3 ) cos ( x ) − 8 sin 3 ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) − 4 sin ( 5 ) sin 2 ( x ) cos ( 3 ) cos ( x ) + 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 4 ( x ) + 4 sin ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) − 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 2 ( x ) + sin ( 3 ) sin ( 5 ) sin ( x ) − 8 sin ( 3 ) cos ( 5 ) cos 5 ( x ) + 8 sin ( 3 ) cos ( 5 ) cos 3 ( x ) − sin ( 3 ) cos ( 5 ) cos ( x )
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
∫ 8 sin ( 5 ) sin 4 ( x ) cos ( 3 ) cos ( x ) d x = 8 sin ( 5 ) cos ( 3 ) ∫ sin 4 ( x ) cos ( x ) d x \int 8 \sin{\left(5 \right)} \sin^{4}{\left(x \right)} \cos{\left(3 \right)} \cos{\left(x \right)}\, dx = 8 \sin{\left(5 \right)} \cos{\left(3 \right)} \int \sin^{4}{\left(x \right)} \cos{\left(x \right)}\, dx ∫ 8 sin ( 5 ) sin 4 ( x ) cos ( 3 ) cos ( x ) d x = 8 sin ( 5 ) cos ( 3 ) ∫ sin 4 ( x ) cos ( x ) d x
Let u = sin ( x ) u = \sin{\left(x \right)} u = sin ( x ) .
Then let d u = cos ( x ) d x du = \cos{\left(x \right)} dx d u = cos ( x ) d x and substitute d u du d u :
∫ u 4 d u \int u^{4}\, du ∫ u 4 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 4 d u = u 5 5 \int u^{4}\, du = \frac{u^{5}}{5} ∫ u 4 d u = 5 u 5
Now substitute u u u back in:
sin 5 ( x ) 5 \frac{\sin^{5}{\left(x \right)}}{5} 5 sin 5 ( x )
So, the result is: 8 sin ( 5 ) sin 5 ( x ) cos ( 3 ) 5 \frac{8 \sin{\left(5 \right)} \sin^{5}{\left(x \right)} \cos{\left(3 \right)}}{5} 5 8 s i n ( 5 ) s i n 5 ( x ) c o s ( 3 )
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − 8 sin 3 ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) ) d x = − 8 cos ( 3 ) cos ( 5 ) ∫ sin 3 ( x ) cos 2 ( x ) d x \int \left(- 8 \sin^{3}{\left(x \right)} \cos{\left(3 \right)} \cos{\left(5 \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 8 \cos{\left(3 \right)} \cos{\left(5 \right)} \int \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}\, dx ∫ ( − 8 sin 3 ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) ) d x = − 8 cos ( 3 ) cos ( 5 ) ∫ sin 3 ( x ) cos 2 ( x ) d x
Rewrite the integrand:
sin 3 ( x ) cos 2 ( x ) = ( 1 − cos 2 ( x ) ) sin ( x ) cos 2 ( x ) \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)} = \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} sin 3 ( x ) cos 2 ( x ) = ( 1 − cos 2 ( x ) ) sin ( x ) cos 2 ( x )
There are multiple ways to do this integral.
Method #1
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute d u du d u :
∫ ( u 4 − u 2 ) d u \int \left(u^{4} - u^{2}\right)\, du ∫ ( u 4 − u 2 ) d u
Integrate term-by-term:
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 4 d u = u 5 5 \int u^{4}\, du = \frac{u^{5}}{5} ∫ u 4 d u = 5 u 5
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 2 ) d u = − ∫ u 2 d u \int \left(- u^{2}\right)\, du = - \int u^{2}\, du ∫ ( − u 2 ) d u = − ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
So, the result is: − u 3 3 - \frac{u^{3}}{3} − 3 u 3
The result is: u 5 5 − u 3 3 \frac{u^{5}}{5} - \frac{u^{3}}{3} 5 u 5 − 3 u 3
Now substitute u u u back in:
cos 5 ( x ) 5 − cos 3 ( x ) 3 \frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3} 5 cos 5 ( x ) − 3 cos 3 ( x )
Method #2
Rewrite the integrand:
( 1 − cos 2 ( x ) ) sin ( x ) cos 2 ( x ) = − sin ( x ) cos 4 ( x ) + sin ( x ) cos 2 ( x ) \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)} ( 1 − cos 2 ( x ) ) sin ( x ) cos 2 ( x ) = − sin ( x ) cos 4 ( x ) + sin ( x ) cos 2 ( x )
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − sin ( x ) cos 4 ( x ) ) d x = − ∫ sin ( x ) cos 4 ( x ) d x \int \left(- \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx ∫ ( − sin ( x ) cos 4 ( x ) ) d x = − ∫ sin ( x ) cos 4 ( x ) d x
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute − d u - du − d u :
∫ u 4 d u \int u^{4}\, du ∫ u 4 d u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 4 ) d u = − ∫ u 4 d u \int \left(- u^{4}\right)\, du = - \int u^{4}\, du ∫ ( − u 4 ) d u = − ∫ u 4 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 4 d u = u 5 5 \int u^{4}\, du = \frac{u^{5}}{5} ∫ u 4 d u = 5 u 5
So, the result is: − u 5 5 - \frac{u^{5}}{5} − 5 u 5
Now substitute u u u back in:
− cos 5 ( x ) 5 - \frac{\cos^{5}{\left(x \right)}}{5} − 5 cos 5 ( x )
So, the result is: cos 5 ( x ) 5 \frac{\cos^{5}{\left(x \right)}}{5} 5 c o s 5 ( x )
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute − d u - du − d u :
∫ u 2 d u \int u^{2}\, du ∫ u 2 d u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 2 ) d u = − ∫ u 2 d u \int \left(- u^{2}\right)\, du = - \int u^{2}\, du ∫ ( − u 2 ) d u = − ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
So, the result is: − u 3 3 - \frac{u^{3}}{3} − 3 u 3
Now substitute u u u back in:
− cos 3 ( x ) 3 - \frac{\cos^{3}{\left(x \right)}}{3} − 3 cos 3 ( x )
The result is: cos 5 ( x ) 5 − cos 3 ( x ) 3 \frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3} 5 c o s 5 ( x ) − 3 c o s 3 ( x )
Method #3
Rewrite the integrand:
( 1 − cos 2 ( x ) ) sin ( x ) cos 2 ( x ) = − sin ( x ) cos 4 ( x ) + sin ( x ) cos 2 ( x ) \left(1 - \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} = - \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \sin{\left(x \right)} \cos^{2}{\left(x \right)} ( 1 − cos 2 ( x ) ) sin ( x ) cos 2 ( x ) = − sin ( x ) cos 4 ( x ) + sin ( x ) cos 2 ( x )
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − sin ( x ) cos 4 ( x ) ) d x = − ∫ sin ( x ) cos 4 ( x ) d x \int \left(- \sin{\left(x \right)} \cos^{4}{\left(x \right)}\right)\, dx = - \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx ∫ ( − sin ( x ) cos 4 ( x ) ) d x = − ∫ sin ( x ) cos 4 ( x ) d x
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute − d u - du − d u :
∫ u 4 d u \int u^{4}\, du ∫ u 4 d u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 4 ) d u = − ∫ u 4 d u \int \left(- u^{4}\right)\, du = - \int u^{4}\, du ∫ ( − u 4 ) d u = − ∫ u 4 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 4 d u = u 5 5 \int u^{4}\, du = \frac{u^{5}}{5} ∫ u 4 d u = 5 u 5
So, the result is: − u 5 5 - \frac{u^{5}}{5} − 5 u 5
Now substitute u u u back in:
− cos 5 ( x ) 5 - \frac{\cos^{5}{\left(x \right)}}{5} − 5 cos 5 ( x )
So, the result is: cos 5 ( x ) 5 \frac{\cos^{5}{\left(x \right)}}{5} 5 c o s 5 ( x )
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute − d u - du − d u :
∫ u 2 d u \int u^{2}\, du ∫ u 2 d u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 2 ) d u = − ∫ u 2 d u \int \left(- u^{2}\right)\, du = - \int u^{2}\, du ∫ ( − u 2 ) d u = − ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
So, the result is: − u 3 3 - \frac{u^{3}}{3} − 3 u 3
Now substitute u u u back in:
− cos 3 ( x ) 3 - \frac{\cos^{3}{\left(x \right)}}{3} − 3 cos 3 ( x )
The result is: cos 5 ( x ) 5 − cos 3 ( x ) 3 \frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3} 5 c o s 5 ( x ) − 3 c o s 3 ( x )
So, the result is: − 8 ( cos 5 ( x ) 5 − cos 3 ( x ) 3 ) cos ( 3 ) cos ( 5 ) - 8 \left(\frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}\right) \cos{\left(3 \right)} \cos{\left(5 \right)} − 8 ( 5 c o s 5 ( x ) − 3 c o s 3 ( x ) ) cos ( 3 ) cos ( 5 )
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − 4 sin ( 5 ) sin 2 ( x ) cos ( 3 ) cos ( x ) ) d x = − 4 sin ( 5 ) cos ( 3 ) ∫ sin 2 ( x ) cos ( x ) d x \int \left(- 4 \sin{\left(5 \right)} \sin^{2}{\left(x \right)} \cos{\left(3 \right)} \cos{\left(x \right)}\right)\, dx = - 4 \sin{\left(5 \right)} \cos{\left(3 \right)} \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx ∫ ( − 4 sin ( 5 ) sin 2 ( x ) cos ( 3 ) cos ( x ) ) d x = − 4 sin ( 5 ) cos ( 3 ) ∫ sin 2 ( x ) cos ( x ) d x
Let u = sin ( x ) u = \sin{\left(x \right)} u = sin ( x ) .
Then let d u = cos ( x ) d x du = \cos{\left(x \right)} dx d u = cos ( x ) d x and substitute d u du d u :
∫ u 2 d u \int u^{2}\, du ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
Now substitute u u u back in:
sin 3 ( x ) 3 \frac{\sin^{3}{\left(x \right)}}{3} 3 sin 3 ( x )
So, the result is: − 4 sin ( 5 ) sin 3 ( x ) cos ( 3 ) 3 - \frac{4 \sin{\left(5 \right)} \sin^{3}{\left(x \right)} \cos{\left(3 \right)}}{3} − 3 4 s i n ( 5 ) s i n 3 ( x ) c o s ( 3 )
The integral of a constant times a function is the constant times the integral of the function:
∫ 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 4 ( x ) d x = 8 sin ( 3 ) sin ( 5 ) ∫ sin ( x ) cos 4 ( x ) d x \int 8 \sin{\left(3 \right)} \sin{\left(5 \right)} \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx = 8 \sin{\left(3 \right)} \sin{\left(5 \right)} \int \sin{\left(x \right)} \cos^{4}{\left(x \right)}\, dx ∫ 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 4 ( x ) d x = 8 sin ( 3 ) sin ( 5 ) ∫ sin ( x ) cos 4 ( x ) d x
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute − d u - du − d u :
∫ u 4 d u \int u^{4}\, du ∫ u 4 d u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 4 ) d u = − ∫ u 4 d u \int \left(- u^{4}\right)\, du = - \int u^{4}\, du ∫ ( − u 4 ) d u = − ∫ u 4 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 4 d u = u 5 5 \int u^{4}\, du = \frac{u^{5}}{5} ∫ u 4 d u = 5 u 5
So, the result is: − u 5 5 - \frac{u^{5}}{5} − 5 u 5
Now substitute u u u back in:
− cos 5 ( x ) 5 - \frac{\cos^{5}{\left(x \right)}}{5} − 5 cos 5 ( x )
So, the result is: − 8 sin ( 3 ) sin ( 5 ) cos 5 ( x ) 5 - \frac{8 \sin{\left(3 \right)} \sin{\left(5 \right)} \cos^{5}{\left(x \right)}}{5} − 5 8 s i n ( 3 ) s i n ( 5 ) c o s 5 ( x )
The integral of a constant times a function is the constant times the integral of the function:
∫ 4 sin ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) d x = 4 cos ( 3 ) cos ( 5 ) ∫ sin ( x ) cos 2 ( x ) d x \int 4 \sin{\left(x \right)} \cos{\left(3 \right)} \cos{\left(5 \right)} \cos^{2}{\left(x \right)}\, dx = 4 \cos{\left(3 \right)} \cos{\left(5 \right)} \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx ∫ 4 sin ( x ) cos ( 3 ) cos ( 5 ) cos 2 ( x ) d x = 4 cos ( 3 ) cos ( 5 ) ∫ sin ( x ) cos 2 ( x ) d x
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute − d u - du − d u :
∫ u 2 d u \int u^{2}\, du ∫ u 2 d u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 2 ) d u = − ∫ u 2 d u \int \left(- u^{2}\right)\, du = - \int u^{2}\, du ∫ ( − u 2 ) d u = − ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
So, the result is: − u 3 3 - \frac{u^{3}}{3} − 3 u 3
Now substitute u u u back in:
− cos 3 ( x ) 3 - \frac{\cos^{3}{\left(x \right)}}{3} − 3 cos 3 ( x )
So, the result is: − 4 cos ( 3 ) cos ( 5 ) cos 3 ( x ) 3 - \frac{4 \cos{\left(3 \right)} \cos{\left(5 \right)} \cos^{3}{\left(x \right)}}{3} − 3 4 c o s ( 3 ) c o s ( 5 ) c o s 3 ( x )
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 2 ( x ) ) d x = − 8 sin ( 3 ) sin ( 5 ) ∫ sin ( x ) cos 2 ( x ) d x \int \left(- 8 \sin{\left(3 \right)} \sin{\left(5 \right)} \sin{\left(x \right)} \cos^{2}{\left(x \right)}\right)\, dx = - 8 \sin{\left(3 \right)} \sin{\left(5 \right)} \int \sin{\left(x \right)} \cos^{2}{\left(x \right)}\, dx ∫ ( − 8 sin ( 3 ) sin ( 5 ) sin ( x ) cos 2 ( x ) ) d x = − 8 sin ( 3 ) sin ( 5 ) ∫ sin ( x ) cos 2 ( x ) d x
Let u = cos ( x ) u = \cos{\left(x \right)} u = cos ( x ) .
Then let d u = − sin ( x ) d x du = - \sin{\left(x \right)} dx d u = − sin ( x ) d x and substitute − d u - du − d u :
∫ u 2 d u \int u^{2}\, du ∫ u 2 d u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 2 ) d u = − ∫ u 2 d u \int \left(- u^{2}\right)\, du = - \int u^{2}\, du ∫ ( − u 2 ) d u = − ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
So, the result is: − u 3 3 - \frac{u^{3}}{3} − 3 u 3
Now substitute u u u back in:
− cos 3 ( x ) 3 - \frac{\cos^{3}{\left(x \right)}}{3} − 3 cos 3 ( x )
So, the result is: 8 sin ( 3 ) sin ( 5 ) cos 3 ( x ) 3 \frac{8 \sin{\left(3 \right)} \sin{\left(5 \right)} \cos^{3}{\left(x \right)}}{3} 3 8 s i n ( 3 ) s i n ( 5 ) c o s 3 ( x )
The integral of a constant times a function is the constant times the integral of the function:
∫ sin ( 3 ) sin ( 5 ) sin ( x ) d x = sin ( 3 ) sin ( 5 ) ∫ sin ( x ) d x \int \sin{\left(3 \right)} \sin{\left(5 \right)} \sin{\left(x \right)}\, dx = \sin{\left(3 \right)} \sin{\left(5 \right)} \int \sin{\left(x \right)}\, dx ∫ sin ( 3 ) sin ( 5 ) sin ( x ) d x = sin ( 3 ) sin ( 5 ) ∫ sin ( x ) d x
The integral of sine is negative cosine:
∫ sin ( x ) d x = − cos ( x ) \int \sin{\left(x \right)}\, dx = - \cos{\left(x \right)} ∫ sin ( x ) d x = − cos ( x )
So, the result is: − sin ( 3 ) sin ( 5 ) cos ( x ) - \sin{\left(3 \right)} \sin{\left(5 \right)} \cos{\left(x \right)} − sin ( 3 ) sin ( 5 ) cos ( x )
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − 8 sin ( 3 ) cos ( 5 ) cos 5 ( x ) ) d x = − 8 sin ( 3 ) cos ( 5 ) ∫ cos 5 ( x ) d x \int \left(- 8 \sin{\left(3 \right)} \cos{\left(5 \right)} \cos^{5}{\left(x \right)}\right)\, dx = - 8 \sin{\left(3 \right)} \cos{\left(5 \right)} \int \cos^{5}{\left(x \right)}\, dx ∫ ( − 8 sin ( 3 ) cos ( 5 ) cos 5 ( x ) ) d x = − 8 sin ( 3 ) cos ( 5 ) ∫ cos 5 ( x ) d x
Rewrite the integrand:
cos 5 ( x ) = ( 1 − sin 2 ( x ) ) 2 cos ( x ) \cos^{5}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} cos 5 ( x ) = ( 1 − sin 2 ( x ) ) 2 cos ( x )
Rewrite the integrand:
( 1 − sin 2 ( x ) ) 2 cos ( x ) = sin 4 ( x ) cos ( x ) − 2 sin 2 ( x ) cos ( x ) + cos ( x ) \left(1 - \sin^{2}{\left(x \right)}\right)^{2} \cos{\left(x \right)} = \sin^{4}{\left(x \right)} \cos{\left(x \right)} - 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + \cos{\left(x \right)} ( 1 − sin 2 ( x ) ) 2 cos ( x ) = sin 4 ( x ) cos ( x ) − 2 sin 2 ( x ) cos ( x ) + cos ( x )
Integrate term-by-term:
Let u = sin ( x ) u = \sin{\left(x \right)} u = sin ( x ) .
Then let d u = cos ( x ) d x du = \cos{\left(x \right)} dx d u = cos ( x ) d x and substitute d u du d u :
∫ u 4 d u \int u^{4}\, du ∫ u 4 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 4 d u = u 5 5 \int u^{4}\, du = \frac{u^{5}}{5} ∫ u 4 d u = 5 u 5
Now substitute u u u back in:
sin 5 ( x ) 5 \frac{\sin^{5}{\left(x \right)}}{5} 5 sin 5 ( x )
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − 2 sin 2 ( x ) cos ( x ) ) d x = − 2 ∫ sin 2 ( x ) cos ( x ) d x \int \left(- 2 \sin^{2}{\left(x \right)} \cos{\left(x \right)}\right)\, dx = - 2 \int \sin^{2}{\left(x \right)} \cos{\left(x \right)}\, dx ∫ ( − 2 sin 2 ( x ) cos ( x ) ) d x = − 2 ∫ sin 2 ( x ) cos ( x ) d x
Let u = sin ( x ) u = \sin{\left(x \right)} u = sin ( x ) .
Then let d u = cos ( x ) d x du = \cos{\left(x \right)} dx d u = cos ( x ) d x and substitute d u du d u :
∫ u 2 d u \int u^{2}\, du ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
Now substitute u u u back in:
sin 3 ( x ) 3 \frac{\sin^{3}{\left(x \right)}}{3} 3 sin 3 ( x )
So, the result is: − 2 sin 3 ( x ) 3 - \frac{2 \sin^{3}{\left(x \right)}}{3} − 3 2 s i n 3 ( x )
The integral of cosine is sine:
∫ cos ( x ) d x = sin ( x ) \int \cos{\left(x \right)}\, dx = \sin{\left(x \right)} ∫ cos ( x ) d x = sin ( x )
The result is: sin 5 ( x ) 5 − 2 sin 3 ( x ) 3 + sin ( x ) \frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)} 5 s i n 5 ( x ) − 3 2 s i n 3 ( x ) + sin ( x )
So, the result is: − 8 ( sin 5 ( x ) 5 − 2 sin 3 ( x ) 3 + sin ( x ) ) sin ( 3 ) cos ( 5 ) - 8 \left(\frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}\right) \sin{\left(3 \right)} \cos{\left(5 \right)} − 8 ( 5 s i n 5 ( x ) − 3 2 s i n 3 ( x ) + sin ( x ) ) sin ( 3 ) cos ( 5 )
The integral of a constant times a function is the constant times the integral of the function:
∫ 8 sin ( 3 ) cos ( 5 ) cos 3 ( x ) d x = 8 sin ( 3 ) cos ( 5 ) ∫ cos 3 ( x ) d x \int 8 \sin{\left(3 \right)} \cos{\left(5 \right)} \cos^{3}{\left(x \right)}\, dx = 8 \sin{\left(3 \right)} \cos{\left(5 \right)} \int \cos^{3}{\left(x \right)}\, dx ∫ 8 sin ( 3 ) cos ( 5 ) cos 3 ( x ) d x = 8 sin ( 3 ) cos ( 5 ) ∫ cos 3 ( x ) d x
Rewrite the integrand:
cos 3 ( x ) = ( 1 − sin 2 ( x ) ) cos ( x ) \cos^{3}{\left(x \right)} = \left(1 - \sin^{2}{\left(x \right)}\right) \cos{\left(x \right)} cos 3 ( x ) = ( 1 − sin 2 ( x ) ) cos ( x )
Let u = sin ( x ) u = \sin{\left(x \right)} u = sin ( x ) .
Then let d u = cos ( x ) d x du = \cos{\left(x \right)} dx d u = cos ( x ) d x and substitute d u du d u :
∫ ( 1 − u 2 ) d u \int \left(1 - u^{2}\right)\, du ∫ ( 1 − u 2 ) d u
Integrate term-by-term:
The integral of a constant is the constant times the variable of integration:
∫ 1 d u = u \int 1\, du = u ∫ 1 d u = u
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − u 2 ) d u = − ∫ u 2 d u \int \left(- u^{2}\right)\, du = - \int u^{2}\, du ∫ ( − u 2 ) d u = − ∫ u 2 d u
The integral of u n u^{n} u n is u n + 1 n + 1 \frac{u^{n + 1}}{n + 1} n + 1 u n + 1 when n ≠ − 1 n \neq -1 n = − 1 :
∫ u 2 d u = u 3 3 \int u^{2}\, du = \frac{u^{3}}{3} ∫ u 2 d u = 3 u 3
So, the result is: − u 3 3 - \frac{u^{3}}{3} − 3 u 3
The result is: − u 3 3 + u - \frac{u^{3}}{3} + u − 3 u 3 + u
Now substitute u u u back in:
− sin 3 ( x ) 3 + sin ( x ) - \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)} − 3 sin 3 ( x ) + sin ( x )
So, the result is: 8 ( − sin 3 ( x ) 3 + sin ( x ) ) sin ( 3 ) cos ( 5 ) 8 \left(- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}\right) \sin{\left(3 \right)} \cos{\left(5 \right)} 8 ( − 3 s i n 3 ( x ) + sin ( x ) ) sin ( 3 ) cos ( 5 )
The integral of a constant times a function is the constant times the integral of the function:
∫ ( − sin ( 3 ) cos ( 5 ) cos ( x ) ) d x = − sin ( 3 ) cos ( 5 ) ∫ cos ( x ) d x \int \left(- \sin{\left(3 \right)} \cos{\left(5 \right)} \cos{\left(x \right)}\right)\, dx = - \sin{\left(3 \right)} \cos{\left(5 \right)} \int \cos{\left(x \right)}\, dx ∫ ( − sin ( 3 ) cos ( 5 ) cos ( x ) ) d x = − sin ( 3 ) cos ( 5 ) ∫ cos ( x ) d x
The integral of cosine is sine:
∫ cos ( x ) d x = sin ( x ) \int \cos{\left(x \right)}\, dx = \sin{\left(x \right)} ∫ cos ( x ) d x = sin ( x )
So, the result is: − sin ( 3 ) sin ( x ) cos ( 5 ) - \sin{\left(3 \right)} \sin{\left(x \right)} \cos{\left(5 \right)} − sin ( 3 ) sin ( x ) cos ( 5 )
The result is: 8 ( − sin 3 ( x ) 3 + sin ( x ) ) sin ( 3 ) cos ( 5 ) − 8 ( cos 5 ( x ) 5 − cos 3 ( x ) 3 ) cos ( 3 ) cos ( 5 ) − 8 ( sin 5 ( x ) 5 − 2 sin 3 ( x ) 3 + sin ( x ) ) sin ( 3 ) cos ( 5 ) + 8 sin ( 5 ) sin 5 ( x ) cos ( 3 ) 5 − 4 sin ( 5 ) sin 3 ( x ) cos ( 3 ) 3 − sin ( 3 ) sin ( x ) cos ( 5 ) − 8 sin ( 3 ) sin ( 5 ) cos 5 ( x ) 5 + 8 sin ( 3 ) sin ( 5 ) cos 3 ( x ) 3 − 4 cos ( 3 ) cos ( 5 ) cos 3 ( x ) 3 − sin ( 3 ) sin ( 5 ) cos ( x ) 8 \left(- \frac{\sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}\right) \sin{\left(3 \right)} \cos{\left(5 \right)} - 8 \left(\frac{\cos^{5}{\left(x \right)}}{5} - \frac{\cos^{3}{\left(x \right)}}{3}\right) \cos{\left(3 \right)} \cos{\left(5 \right)} - 8 \left(\frac{\sin^{5}{\left(x \right)}}{5} - \frac{2 \sin^{3}{\left(x \right)}}{3} + \sin{\left(x \right)}\right) \sin{\left(3 \right)} \cos{\left(5 \right)} + \frac{8 \sin{\left(5 \right)} \sin^{5}{\left(x \right)} \cos{\left(3 \right)}}{5} - \frac{4 \sin{\left(5 \right)} \sin^{3}{\left(x \right)} \cos{\left(3 \right)}}{3} - \sin{\left(3 \right)} \sin{\left(x \right)} \cos{\left(5 \right)} - \frac{8 \sin{\left(3 \right)} \sin{\left(5 \right)} \cos^{5}{\left(x \right)}}{5} + \frac{8 \sin{\left(3 \right)} \sin{\left(5 \right)} \cos^{3}{\left(x \right)}}{3} - \frac{4 \cos{\left(3 \right)} \cos{\left(5 \right)} \cos^{3}{\left(x \right)}}{3} - \sin{\left(3 \right)} \sin{\left(5 \right)} \cos{\left(x \right)} 8 ( − 3 s i n 3 ( x ) + sin ( x ) ) sin ( 3 ) cos ( 5 ) − 8 ( 5 c o s 5 ( x ) − 3 c o s 3 ( x ) ) cos ( 3 ) cos ( 5 ) − 8 ( 5 s i n 5 ( x ) − 3 2 s i n 3 ( x ) + sin ( x ) ) sin ( 3 ) cos ( 5 ) + 5 8 s i n ( 5 ) s i n 5 ( x ) c o s ( 3 ) − 3 4 s i n ( 5 ) s i n 3 ( x ) c o s ( 3 ) − sin ( 3 ) sin ( x ) cos ( 5 ) − 5 8 s i n ( 3 ) s i n ( 5 ) c o s 5 ( x ) + 3 8 s i n ( 3 ) s i n ( 5 ) c o s 3 ( x ) − 3 4 c o s ( 3 ) c o s ( 5 ) c o s 3 ( x ) − sin ( 3 ) sin ( 5 ) cos ( x )
Now simplify:
− 2 cos 2 ( 2 x ) cos ( x + 2 ) 5 − 4 cos ( 2 x ) cos ( x − 2 ) 5 + 3 cos ( x + 2 ) 5 − cos ( 3 x − 8 ) 6 + cos ( 3 x − 2 ) 2 - \frac{2 \cos^{2}{\left(2 x \right)} \cos{\left(x + 2 \right)}}{5} - \frac{4 \cos{\left(2 x \right)} \cos{\left(x - 2 \right)}}{5} + \frac{3 \cos{\left(x + 2 \right)}}{5} - \frac{\cos{\left(3 x - 8 \right)}}{6} + \frac{\cos{\left(3 x - 2 \right)}}{2} − 5 2 cos 2 ( 2 x ) cos ( x + 2 ) − 5 4 cos ( 2 x ) cos ( x − 2 ) + 5 3 cos ( x + 2 ) − 6 cos ( 3 x − 8 ) + 2 cos ( 3 x − 2 )
Add the constant of integration:
− 2 cos 2 ( 2 x ) cos ( x + 2 ) 5 − 4 cos ( 2 x ) cos ( x − 2 ) 5 + 3 cos ( x + 2 ) 5 − cos ( 3 x − 8 ) 6 + cos ( 3 x − 2 ) 2 + c o n s t a n t - \frac{2 \cos^{2}{\left(2 x \right)} \cos{\left(x + 2 \right)}}{5} - \frac{4 \cos{\left(2 x \right)} \cos{\left(x - 2 \right)}}{5} + \frac{3 \cos{\left(x + 2 \right)}}{5} - \frac{\cos{\left(3 x - 8 \right)}}{6} + \frac{\cos{\left(3 x - 2 \right)}}{2}+ \mathrm{constant} − 5 2 cos 2 ( 2 x ) cos ( x + 2 ) − 5 4 cos ( 2 x ) cos ( x − 2 ) + 5 3 cos ( x + 2 ) − 6 cos ( 3 x − 8 ) + 2 cos ( 3 x − 2 ) + constant
The answer is:
− 2 cos 2 ( 2 x ) cos ( x + 2 ) 5 − 4 cos ( 2 x ) cos ( x − 2 ) 5 + 3 cos ( x + 2 ) 5 − cos ( 3 x − 8 ) 6 + cos ( 3 x − 2 ) 2 + c o n s t a n t - \frac{2 \cos^{2}{\left(2 x \right)} \cos{\left(x + 2 \right)}}{5} - \frac{4 \cos{\left(2 x \right)} \cos{\left(x - 2 \right)}}{5} + \frac{3 \cos{\left(x + 2 \right)}}{5} - \frac{\cos{\left(3 x - 8 \right)}}{6} + \frac{\cos{\left(3 x - 2 \right)}}{2}+ \mathrm{constant} − 5 2 cos 2 ( 2 x ) cos ( x + 2 ) − 5 4 cos ( 2 x ) cos ( x − 2 ) + 5 3 cos ( x + 2 ) − 6 cos ( 3 x − 8 ) + 2 cos ( 3 x − 2 ) + constant
The answer (Indefinite)
[src]
/ / 3 5 \ / 3 5 \ / 3 \ 5 3 3 3 5
| | cos (x) cos (x)| | 2*sin (x) sin (x) | | sin (x) | 8*cos (x)*sin(3)*sin(5) 4*cos (x)*cos(3)*cos(5) 4*sin (x)*cos(3)*sin(5) 8*cos (x)*sin(3)*sin(5) 8*sin (x)*cos(3)*sin(5)
| sin(4*x - 3)*cos(x + 5) dx = C - cos(5)*sin(3)*sin(x) - cos(x)*sin(3)*sin(5) - 8*|- ------- + -------|*cos(3)*cos(5) - 8*|- --------- + ------- + sin(x)|*cos(5)*sin(3) + 8*|- ------- + sin(x)|*cos(5)*sin(3) - ----------------------- - ----------------------- - ----------------------- + ----------------------- + -----------------------
| \ 3 5 / \ 3 5 / \ 3 / 5 3 3 3 5
/
− cos ( 5 x + 2 ) 10 − cos ( 3 x − 8 ) 6 -{{\cos \left(5\,x+2\right)}\over{10}}-{{\cos \left(3\,x-8\right)
}\over{6}} − 10 cos ( 5 x + 2 ) − 6 cos ( 3 x − 8 )
The graph
0.00 1.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 2 -2
4*cos(1)*cos(6) sin(1)*sin(6) sin(3)*sin(5) 4*cos(3)*cos(5)
- --------------- - ------------- - ------------- + ---------------
15 15 15 15
5 cos 8 + 3 cos 2 30 − 3 cos 7 + 5 cos 5 30 {{5\,\cos 8+3\,\cos 2}\over{30}}-{{3\,\cos 7+5\,\cos 5}\over{30}} 30 5 cos 8 + 3 cos 2 − 30 3 cos 7 + 5 cos 5
=
4*cos(1)*cos(6) sin(1)*sin(6) sin(3)*sin(5) 4*cos(3)*cos(5)
- --------------- - ------------- - ------------- + ---------------
15 15 15 15
− 4 cos ( 1 ) cos ( 6 ) 15 + 4 cos ( 3 ) cos ( 5 ) 15 − sin ( 3 ) sin ( 5 ) 15 − sin ( 1 ) sin ( 6 ) 15 - \frac{4 \cos{\left(1 \right)} \cos{\left(6 \right)}}{15} + \frac{4 \cos{\left(3 \right)} \cos{\left(5 \right)}}{15} - \frac{\sin{\left(3 \right)} \sin{\left(5 \right)}}{15} - \frac{\sin{\left(1 \right)} \sin{\left(6 \right)}}{15} − 15 4 cos ( 1 ) cos ( 6 ) + 15 4 cos ( 3 ) cos ( 5 ) − 15 sin ( 3 ) sin ( 5 ) − 15 sin ( 1 ) sin ( 6 )
Use the examples entering the upper and lower limits of integration.